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Basic idea For a long time I was looking for a numerical system that would allow to compare infinite sets. In contrast to Cantor's approach that empathizes the possibility of on-to-one correspondence between sets, I want that adding an element to even an infinite set should increase its "quantity" and removing an element should decrease. For instance, 1,2,3,4,5... should have greater quantity than 1,2,4,5... Similarly I want more dense sets having greater quantity, that is, 1,2,3,4,5... having greater quantity than 1,3,5,7... and slmaller than 1,3/2,2,5/2,3,... This way I came to the following considerations. First of all, we extend the real numbers with non-standard numbers. Each non-standard number consists of a standard part and non-standard part. The numbers whose standard part is zero we call pure non-standard. For instance, if p''' is a pure non-standard number, then '''p+1 has standard part 1'''. Now we introduce the notion of 'quantity' of a subset of real numbers '''q(S). - If the set of reals is finite, then the quantity of that set is equal to the number of its members. - The quantity of all integers we designate as \Omega=2\tau . It is a pure non-standard number. - If two sets differ by only the presence or absence of finite number of elements then the non-standard parts of their quantities are equal. - If two sets differ by only the position of finite number of elements, their quantities are equal. - For non-intersecting sets S_1 and S_2 , q(S_1\cup S_2)=q(S_1)+q(S_2) - Quantities of sets symmetric against zero are equal. - Quantities of uniformly distributed sets are proportional to their densities Given these properties, lets find the quantity of the natural numbers q(\mathbb{N}) . We know that \mathbb{Z}=\{-1,-2,-3,...\}\cup\{0\}\cup\{1,2,3,...\}=\mathbb{N^-}\cup\{0\}\cup\mathbb{N} . Now q(\{0\})=1 (it is a finite set) and q(\mathbb{N^-})=q(\mathbb{N}) . So, \Omega=2q(\mathbb{N})+1 . We designate q(\mathbb{N}) as \omega_- , so \omega_-=\frac12 \Omega-\frac12=\tau-1/2 . It is not a pure non-standard number, its standard part is -1/2 . The quantity of all non-negative integers is greater by one, so we designate it \omega_+=\omega_-+1=\tau+1/2 Here are some other examples: - The quantity of even numbers is equal to the quantity of odd numbers, is equal to \Omega/2=\tau - The quantity of the numbers of the form \frac{2n-1}2 with natural n (1/2, 3/2, 5/2,...) is \frac{\Omega}2=\tau - The quantity of positive even numbers is \tau/2 , the quantity of positive odd numbers is \tau/2-1/2 , the quantity of non-negative odd numbers is \tau/2+1/2 . - The quantity of complex integers ordered lexicographically is \Omega^2=4\tau^2 Some sets and their quantities: Definition using series Another problem that for a long time interested me was the meaning of generalized sums of diverging series. I was looking for a non-Archimedian number system that would give the meaning to those sums. And now it seems that Ramanujan's summation of diverging series finally got its place. Now we define that to any divergent series there corresponds a non-standard number. The standard part of that number is given by the Ramanujan's sunmmation of the series. That way we see that \operatorname{st} q(\mathbb{N})= \operatorname{st} \omega_-=\sum_{n\ge1}^{\Re}1=-1/2 Exponentiation of non-standard numbers Examining the Faulhaber's formula for Ramanujan's summation one can come to the following striking insight on the exponentiation of non-standard numbers. : \operatorname{st}\omega_-^n=B_n : \operatorname{st}\omega_+^n=B^'_n Where B_n are the first Bernoulli numbers and B^'_n are the second Bernoulli numbers. Indeed, we can see that \operatorname{st}\omega_-=-1/2, \operatorname{st}\omega_+=1/2, \operatorname{st}\omega_-^2=1/6, \operatorname{st}\omega_-^3=0 etc. Given that Bernoulli numbers can be expressed through Hurwitz Zeta function, we can generalize: : \operatorname{st}\omega_-^x=-x\zeta(1-x,0) : \operatorname{st}\omega_+^x=-x\zeta(1-x,1)=-x\zeta(1-x) This allows to represent zeta function in exponential form: : (x-1)\zeta(x)= \operatorname{st}\omega_-^{1-x} or, more generally, : \operatorname{st}(\omega_-+z)^n= B_n(z) : \operatorname{st}(\tau+y)^x=-x\zeta(1-x,1/2+y) Moreover, now any series containing Bernoulli numbers can be represented as power series over non-standard numbers. From the Riemann functional equation it follows: \operatorname{st}\omega_+^{-x}=\operatorname{st}\frac{-\omega_+^{x+1} 2^x\pi^{x+1}}{\sin(\pi x/2)\Gamma(x)(x+1)} \tau^x = \omega_+^x (2^(1 - x) - 1) Expression for derivative If f(x) is Newton-analytic (equal to its Newton series), the following holds: : f'(x)=\operatorname{st}(f(\omega_++x)-f(\omega_-+x))=\operatorname{st} \Delta f(\omega_-+x) : f'(x)=\operatorname{st}(f(\omega_++x)-f(-\omega_++x)) : f'(x)=\operatorname{st}(f(-\omega_-+x)-f(\omega_-+x)) If f(x) is odd, : \operatorname{st}f(\omega_+)=-\operatorname{st}f(\omega_-)= \frac12 f'(0) Improper integrals Integrals can be transformed into sets of weighted dots using the following principle: an integral over an interval (a_i,b_i) can be replaced with a weighted dot at the center of mass of the figure under the graphic of the integrated function, with weight of the figure's area: The x coordinate of the weighted dot will be x_i=\frac{\int_{a_i}^{b_i} x f(x) dx}{\int_{a_i}^{b_i} f(x) dx} the weight is : p_i=\int_{a_i}^{b_i} f(x) dx Now, if we can represent an infinite integral with a set of weighted dots located in positive integers, the corresponding series would be : \sum_{i=1}^\infty p_i One method that uses the above principle is as follows. Given the function f(x) , : p_k=\frac{\int_k^{k+1} x f(x) \, ds}{\int_k^{k+1} f(x) \, dx} : \int_0^\infty f(x)=\int_0^{p(0)} f(x) \, dx+\sum _{k=1}^{\infty } \int_{p(k-1)}^{p(k)} f(x) \, dx For analytic functions, the following holds: : \operatorname{st}\int_0^\infty f(x) dx = \sum_{n=1}^{\infty} \frac{f^{(n-1)} (0)}{n!} B_n(1/2) Consequences If f(x) is periodic and integral over period is zero, then : \int_{-\infty}^{+\infty} f(x) dx=0 If such function is also even, then : \int_0^{+\infty} f(x) dx=0 Norm One can define the norm of non-standard numbers, by analogy with complex numbers: : \|w\|=\exp(\operatorname{st}\ln w) If so, the following holds: : \|\omega_+\|=e^{-\gamma} Standard parts of some expressions Given the above definitions, we have a lot of relations between trigonometric functions, for instance, : \operatorname{st} \cos (z\omega_-)=\operatorname{st} \cos (z\omega_+)=\frac z2 \cot \left(\frac z2\right) : \operatorname{st} \cosh (z\omega_-)=\operatorname{st} \cosh (z\omega_+)=\frac z2 \coth \left(\frac z2\right) : \operatorname{st} e^{z\omega_-}=\frac{z}{e^{z}-1} : \operatorname{st} e^{z\tau}=\frac{z}{2} \operatorname{csch}\left(\frac{z}{2}\right) : \operatorname{st} \left(\frac{1}{\pi^2 \tau+\pi x}+\frac{1}{\pi^2 \tau-\pi x}\right)=(\sec x)^2 : \operatorname{st}\ln (\omega_-+z)=\psi(z) Particularly, : \operatorname{st}\ln \omega_+=-\gamma : \operatorname{st}\frac1{\pi }\ln \left(\frac{\omega _-+\frac{z}{\pi }}{\omega _+-\frac{z}{\pi }}\right)=-\cot (z) : \operatorname{st} \frac1\pi\ln \left(\frac{\tau +\frac{z}{\pi }}{\tau -\frac{z}{\pi }}\right)=\tan (z) : \operatorname{st}\sin (\omega_-+x) = \frac{1}{2} \cot \left(\frac{1}{2}\right) \sin x -\frac{1}{2} \cos x Particularly, : \operatorname{st}\sin \omega_-=-1/2 , : \operatorname{st}\sin \omega_+=1/2 , : \operatorname{st}\sin \tau=0 : \operatorname{st}\cos (\omega_-+x) = \frac{1}{2} \csc \left(\frac{1}{2}\right) \cos \left(\frac{1}{2}- x \right) : \operatorname{st}\sin (a\omega_-+x) = \frac{a}{2} \cot \left(\frac{a}{2}\right) \sin x -\frac{a}{2} \cos x : \operatorname{st}\cos (a\omega_-+x) = \frac{a}{2} \csc \left(\frac{a}{2}\right) \cos \left(\frac{a}{2}- x\right) : \operatorname{st}\cos (\pi\tau+x)=-\frac\pi{2}\cos x : \operatorname{st}\sin (\pi\tau+x)=-\frac\pi{2}\sin x : \operatorname{st}\frac{(\omega_-+x)^{\omega_-+x}}{\omega_-^{\omega_-}}=\Gamma(x+1) Identities : \sum_{k=0}^\infty 1=\omega_+ : \sum_{k=1}^\infty 1=\omega_- : \sum_{k=0}^\infty k^n=\frac{\omega_-^{n+1}}{n+1} ( n\ge 1 ) : \sum_{k=0}^\infty (k+1)^n=\frac{\omega_+^{n+1}}{n+1} : \sum_{k=0}^\infty k=\sum_{k=1}^\infty k=\frac{\tau^2}2-\frac1{24} : \sum_{k=0}^\infty (-1)^k=\frac12 : \int_0^\infty dx = \omega_-+1/2=\tau : \int_0^\infty x\, dx =\frac{\tau^2}2 : \int_0^\infty x^n\, dx =\frac{\tau^{n+1}}{n+1} : \int_0^\infty \cos x\, dx =0